Modifying the usual Bayesian update with a learning rate \(\eta \)  as in (REF ) is a simple change, but it has some unexpected consequences. In particular, [1]} showed that, for a correctly specified model, consistency of the generalized Bayes posterior \(\Pi _n^{(\eta )}\)  in (REF ) could be established for any \(\eta < 1\) , with only local conditions on the prior—as opposed to the local and global conditions required for consistency with \(\eta =1\)  [2]}, [3]}. The intuition given by [4]} is that inconsistencies result from the posterior over-fitting or tracking the data too closely, and the fractional power discounts the data slight to prevent this over-fitting. The Walker–Hjort result has been extended to cover posterior concentration rates, where the removal of the global prior conditions—usually formulated in terms of metric entropy [5]}, [6]}—leads to simpler proofs and generally (at least slightly) faster rates. See [7]} for one of the first papers exploring these ideas, and [8]} and [9]} for more recent contributions. The fractional power has also been employed recently in work on high-dimensional problems using an empirical or data-driven prior [10]}, [11]}, [12]} where, again, the fractional power is motivated by the desire to prevent over-fitting; see, also, [13]} and [14]} for some potential benefits of \(\eta < 1\)  to uncertainty quantification.
